cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-5 of 5 results.

A322356 Product of such primes p that both p and p-2 divide n, and p-2 is also prime.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 5, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 5, 1, 1, 1, 1, 7, 1, 1, 1, 1, 1, 1, 1, 1, 1, 5, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 5, 1, 1, 1, 1, 1, 1, 1, 1, 1, 7, 1, 1, 1, 1, 5, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 5, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 35
Offset: 1

Views

Author

Antti Karttunen, Dec 16 2018

Keywords

Comments

Product of those distinct greater twin primes (A006512) that divide n for which the corresponding lesser twin prime (A001359) also divides n.

Examples

			For n = 105 = 3*5*7, a(105) = 5*7 = 35.

Links

Crossrefs

Cf. A001359, A006512, A322354, A322357, A322358.

Programs

  • Mathematica
    f[p_, n_] := If[PrimeQ[p + 2] && Divisible[n, p + 2], p + 2, 1]; a[n_] := Times @@ (f[#, n] & /@ FactorInteger[n][[;; , 1]]); Array[a, 120] (* Amiram Eldar, Dec 16 2018 *)
  • PARI
    A322356(n) = { my(f = factor(n), m=1); for(i=1, #f~, if(isprime(f[i,1]+2)&&!(n%(f[i,1]+2)), m *= (f[i,1]+2))); (m); };

Formula

a(n) = A322354(n) / A322357(n).
A001221(a(n)) = A001222(a(n)) = A322358(n).

A322976 Number of divisors d of n such that d+2 is prime.

Original entry on oeis.org

1, 1, 2, 1, 2, 2, 1, 1, 3, 2, 2, 2, 1, 1, 4, 1, 2, 3, 1, 2, 3, 2, 1, 2, 2, 1, 4, 1, 2, 4, 1, 1, 3, 2, 3, 3, 1, 1, 3, 2, 2, 3, 1, 2, 6, 1, 1, 2, 1, 2, 4, 1, 1, 4, 3, 1, 3, 2, 2, 4, 1, 1, 4, 1, 3, 3, 1, 2, 3, 3, 2, 3, 1, 1, 4, 1, 3, 3, 1, 2, 5, 2, 1, 3, 3, 1, 4, 2, 1, 6, 1, 1, 2, 1, 3, 2, 1, 1, 5, 2, 2, 4, 1, 1, 7
Offset: 1

Views

Author

Antti Karttunen, Jan 04 2019

Keywords

Examples

			10395 has 32 divisors: [1, 3, 5, 7, 9, 11, 15, 21, 27, 33, 35, 45, 55, 63, 77, 99, 105, 135, 165, 189, 231, 297, 315, 385, 495, 693, 945, 1155, 1485, 2079, 3465, 10395]. When 2 is added to each, as 1+2 = 3, 3+2 = 5, 5+2 = 7, etc, the only sums that are primes are: [3, 5, 7, 11, 13, 17, 23, 29, 37, 47, 79, 101, 107, 137, 167, 191, 233, 317, 947, 1487, 2081, 3467], thus (a10395) = 22.

Links

Crossrefs

Cf. A010051, A067513, A072627, A100821, A322358, A322975, A322977, A322978.

Programs

  • Mathematica
    Array[DivisorSum[#, 1 &, PrimeQ[# + 2] &] &, 105] (* Michael De Vlieger, Jan 04 2019 *)
  • PARI
    A322976(n) = sumdiv(n, d, isprime(d+2));

Formula

a(n) = Sum_{d|n} A010051(d+2).
a(A000040(n)) = 1 + A100821(n).

A322975 Number of divisors d of n such that d-2 is prime.

Original entry on oeis.org

0, 0, 0, 1, 1, 0, 1, 1, 1, 1, 0, 1, 1, 1, 2, 1, 0, 1, 1, 2, 2, 0, 0, 1, 2, 1, 1, 2, 0, 2, 1, 1, 1, 0, 2, 2, 0, 1, 2, 2, 0, 2, 1, 1, 4, 0, 0, 1, 2, 2, 0, 2, 0, 1, 2, 2, 1, 0, 0, 3, 1, 1, 4, 1, 2, 1, 0, 1, 1, 2, 0, 2, 1, 0, 4, 2, 1, 2, 0, 2, 2, 0, 0, 3, 2, 1, 0, 1, 0, 4, 3, 1, 1, 0, 2, 1, 0, 2, 3, 3, 0, 0, 1, 2, 5
Offset: 1

Views

Author

Antti Karttunen, Jan 04 2019

Keywords

Examples

			10395 has 32 divisors: [1, 3, 5, 7, 9, 11, 15, 21, 27, 33, 35, 45, 55, 63, 77, 99, 105, 135, 165, 189, 231, 297, 315, 385, 495, 693, 945, 1155, 1485, 2079, 3465, 10395]. When 2 is subtracted from each, as 1-2 = -1, 3-2 = 1, 5-2 = 3, etc, the only differences that are primes are: [3, 5, 7, 13, 19, 31, 43, 53, 61, 97, 103, 163, 229, 313, 383, 691, 1153, 1483, 3463], thus (a10395) = 19.

Links

Crossrefs

Cf. A010051, A062301, A067513, A072627, A322358, A322976, A322977, A322978.

Programs

  • Mathematica
    Array[DivisorSum[#, 1 &, PrimeQ[# - 2] &] &, 105] (* Michael De Vlieger, Jan 04 2019 *)
  • PARI
    A322975(n) = sumdiv(n, d, isprime(d-2));

Formula

a(n) = Sum_{d|n, d>2} A010051(d-2).
a(A000040(n)) = A062301(n).

A323068 Number of divisors d of n such that A049820(d) > 0 and A049820(d) is also a divisor of n.

Original entry on oeis.org

0, 0, 1, 1, 0, 2, 0, 2, 1, 0, 0, 4, 0, 0, 2, 2, 0, 3, 0, 1, 1, 0, 0, 5, 0, 0, 1, 1, 0, 4, 0, 2, 1, 0, 1, 6, 0, 0, 1, 2, 0, 2, 0, 1, 2, 0, 0, 6, 0, 0, 1, 1, 0, 3, 0, 2, 1, 0, 0, 6, 0, 0, 1, 2, 0, 2, 0, 1, 1, 2, 0, 7, 0, 0, 2, 1, 0, 2, 0, 2, 1, 0, 0, 4, 0, 0, 1, 2, 0, 5, 0, 1, 1, 0, 0, 6, 0, 0, 2, 1, 0, 2, 0, 2, 3
Offset: 1

Views

Author

Antti Karttunen, Jan 05 2019

Keywords

Comments

Records 0, 1, 2, 4, 5, 6, 7, 8, 9, 10, 12, 13, 14, 15, 16, 19, 20, 22, 27, 30, ... occur at n = 1, 3, 6, 12, 24, 36, 72, 144, 240, 360, 720, 1440, 1680, 2640, 3360, 5040, 7920, 10080, 30240, 55440, ...

Links

Crossrefs

Cf. A000005, A049820, A322358, A323069.

Programs

  • PARI
    A323068(n) = sumdiv(n,d,my(t=(d-numdiv(d))); ((t>0)&&!(n%t)));

Formula

Sum_{d|n} [A049820(d) > 0 and A049820(d)|n], where [ ] is the Iverson bracket.
a(n) >= A323069(n) => A322358(n).

A323069 Number of divisors d of n such that A049820(d) > 1 and A049820(d) is also a divisor of n.

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 2, 0, 0, 1, 1, 0, 2, 0, 0, 0, 0, 0, 3, 0, 0, 0, 0, 0, 3, 0, 1, 0, 0, 1, 4, 0, 0, 0, 1, 0, 1, 0, 0, 1, 0, 0, 4, 0, 0, 0, 0, 0, 2, 0, 1, 0, 0, 0, 4, 0, 0, 0, 1, 0, 1, 0, 0, 0, 2, 0, 5, 0, 0, 1, 0, 0, 1, 0, 1, 0, 0, 0, 2, 0, 0, 0, 1, 0, 4, 0, 0, 0, 0, 0, 4, 0, 0, 1, 0, 0, 1, 0, 1, 2
Offset: 1

Views

Author

Antti Karttunen, Jan 05 2019

Keywords

Links

Crossrefs

Cf. A000005, A049820, A322358, A323068.

Programs

  • PARI
    A323069(n) = sumdiv(n,d,my(t=(d-numdiv(d))); ((t>1)&&!(n%t)));

Formula

Sum_{d|n} [A049820(d) > 1 and A049820(d)|n], where [ ] is the Iverson bracket.
a(n) <= A323068(n).
a(n) >= A322358(n).
Showing 1-5 of 5 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement.